Editing in Adobe Camera RAW vs Photoshop.

They won’t be mathematically identical, for a number of reasons. 

• Camera Raw/Lightroom start from the raw file, not yet rendered to channels such as RGB. The editing algorithms are optimized for that type of source, camera raw data. When they render an edited image to be passed on to Photoshop, they base the render on the current Workspace settings (color space, bit depth…) in the Workflow tab of Camera Raw preferences. 
• Photoshop starts from images already rendered to channels in a color space such as sRGB or a specific form of CMYK, basing adjustments on that document’s color space/bit depth settings. The algorithms are optimized for rendered (not raw) images, so they don’t work exactly the same way as in Camera Raw/Lightroom. If you open a raw file in Photoshop, it's first rendered through the current Camera Raw settings, renders that to the color space settings, and then Photoshop starts from that. So you could have different starting points even if the edits were the same. 

But the edits won't be the same. You said “increase brightness by 12%." Well, there is no Brightness adjustment in Camera Raw. There used to be, but in 2012 the Brightness algorithm was replaced by the more advanced Exposure option. So if you wanted to match the Camera Raw Exposure option in Photoshop, you might have to use the Exposure command in Photoshop, not Brightness. I think both Exposure controls assume a linear (not gamma-corrected) source, but I'm not sure.

In Photoshop, the Brightness adjustment is a very simple move. (It used to be even simpler, back when all it did was shift all tones up or down by a fixed value, which allowed unwanted clipping. You can still make it work the old way if you enable the Legacy option in Brightness.) In Camera Raw, the Exposure control has non-linear adjustments built into it to protect and optimized image quality, especially in the highlights. I am not sure if the Exposure adjustment in Photoshop is that sophisticated.

If that sounds complicated and confusing, the details aren’t important, what is important is that once you say one image will be edited raw and the other edited in Photoshop, you branch off the two images into two very different processing pipelines that are not built the same way at all, with many ways for them to be different. That’s on top of also having to make sure that the baseline image settings (color space, bit depth…) are set the same way for both images you’re comparing.

One difference you can count on is that edits done in raw are potentially higher quality than edits done after a raw image is converted to RGB/CMYK, because raw preserves all of the original image data/quality. So in theory it’s better to take an image as far as you can in Camera Raw before converting it to an RGB/CMYK file for Photoshop Now, that doesn’t mean editing in Photoshop is always visibly worse. If image quality of a raw photo is in good shape and you export it to Photoshop as 16 or 32 bits per channel, quality and editing latitude will be almost as good as a raw file. There will be some loss, but it might not be important or visible depending on how far you want to push the edits.

The other important takeaway is that if you understand the principles of preserving image quality, and you understand how to best use both sets of editing controls (Camera Raw and Photoshop), you can get sufficiently high-quality JPEGs out of Camera Raw or Photoshop. Or at least any differences won’t be enough to disqualify one or the other. An image edited as identically as possible in each might not be mathematically identical, but you might be able to make them visually indistinguishable.

What can help you decide which tool to use is what a specific image needs. With some images, I look at it and everything it needs can be done in Camera Raw (or done better/more easily than in Photoshop) and I can just export from there and be done. With other images, Camera Raw will be out of its depth and I really should send it to Photoshop to get it right. But again, either way, both sets of tools, if used appropriately, can export JPEGs of the quality you need.